Root Systems & Clifford Algebras

Dechant, Pierre-Philippe ORCID: https://orcid.org/0000-0002-4694-4010 (2017) Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence. In: Pure Mathematics Colloquium, 13th January 2017, University of St Andrews, Scotland. (Unpublished)

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RaY Research at the University of York St John For more information please contact RaY at [email protected] Viruses, root systems and aﬃne extensions Cliﬀord algebras and exceptional root systems The Coxeter plane

Root systems & Cliﬀord algebras: from symmetries of viruses to E8 & an ADE correspondence

Pierre-Philippe Dechant

Departments of Mathematics and Biology, York Centre for Complex Systems Analysis, University of York

St Andrews Pure Maths Colloquium – January 13, 2017

Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Clifford algebras and exceptional root systems The Coxeter plane Main results

New affine symmetry principle for viruses and fullerenes H3 (icosahedral symmetry) induces the E8 root system Each 3D root system induces a 4D root system This correspondence extends to exponents in the Coxeter plane (not just the original Trinity) and ADE Lie algebras Clifford algebra is a very natural framework for root systems and reflection groups in general

Pierre-Philippe Dechant Root systems & Cliﬀord algebras: from symmetries of viruses to E8 & an ADE correspondence Overview

1 Viruses, root systems and aﬃne extensions Viruses Root systems Aﬃne extensions

2 Cliﬀord algebras and exceptional root systems Cliﬀord basics E8 from the icosahedron 3D to 4D spinor induction Trinities and McKay correspondence

3 The Coxeter plane Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions The Icosahedron

Rotational icosahedral group is I = A5 of order 60

Full icosahedral group is the Coxeter group H3 of order 120 (including reﬂections/inversion); generated by the root system icosidodecahedron

Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Icosahedral viruses

Two viral surface layouts: a T = 4 Caspar-Klug quasiequivalent triangulation, and a pseudo T = 7 defying Caspar-Klug theory, which is based on a kite-rhombus tiling instead (HPV)

Root system Φ: set of α2 α1 + α2 vectors α in a vector space with an inner product such that 1. Φ ∩ α = {−α,α} ∀ α ∈ Φ −α1 α1 R 2. sα Φ = Φ ∀ α ∈ Φ Simple roots: express −(α1 + α2) −α2 every element of Φ via a Z-linear combination. (v|α) reﬂection/Coxeter groups s : v → s (v) = v − 2 α α α (α|α)

(αi |αj ) |αj | Cartan matrix of αi s is Aij = 2 = 2 cosθij (αi |αi ) |αi | 2 −1 A : A = 2 −1 2 Coxeter-Dynkin diagrams: node = simple root, no link = roots π π orthogonal, simple link = roots at 3 , link with label m = angle m . 4 5 n A3 B3 H3 I2(n)

Lie group: manifold of continuous symmetries (gauge theories, spacetime) Lie algebra: inﬁnitesimal version near the identity Non-trivial part is given by a root lattice Weyl group is a crystallographic Coxeter group: An,Bn/Cn,Dn,G2,F4,E6,E7,E8 generated by a root system. So via this route root systems are always crystallographic. Neglect non-crystallographic root systems I2(n),H3,H4 .

An affine Coxeter group is the extension of a Coxeter group by an aff affine reflection in a hyperplane not containing the origin sα0 whose geometric action is given by

aﬀ 2(α0|v) sα0 v = α0 + v − α0 (α0|α0)

Non-distance preserving: includes the translation generator aﬀ Tv = v + α0 = sα0 sα0 v

Aﬃne extensions – A2

Aﬃne extensions of crystallographic Coxeter groups lead to a tessellation of the plane and a lattice.

Non-crystallographic Coxeter groups H2 ⊂ H3 ⊂ H4 τ(α1 + α2) α1 + τα2 τα + α α2 1 2

−α1 α1

−(τα1 + α2) −α2

−(α1 + τα2) −τ(α1 + α2) H2 ⊂ H3 ⊂ H4: 10, 120, 14,400 elements, the only Coxeter groups that generate rotational symmetries of order 5 linear combinations now in the extended integer ring 1 √ π [τ] = {a + τb|a,b ∈ } golden ratio τ = (1 + 5) = 2cos Z Z 2 5 1 √ 2π x2 = x + 1 τ0 = σ = (1 − 5) = 2cos τ + σ = 1,τσ = −1 2 5

Pierre-Philippe Dechant Root systems & Cliﬀord algebras: from symmetries of viruses to E8 & an ADE correspondence G

A random translation would give 5 secondary pentagons, i.e. 25 points. Here we have degeneracies due to ‘coinciding points’.

Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Affine extensions of non-crystallographic root systems?

Unit translation along a vertex of a unit pentagon

Pierre-Philippe Dechant Root systems & Cliﬀord algebras: from symmetries of viruses to E8 & an ADE correspondence G

A random translation would give 5 secondary pentagons, i.e. 25 points. Here we have degeneracies due to ‘coinciding points’.

Unit translation along a vertex of a unit pentagon

G T

Unit translation along a vertex of a unit pentagon

G G T

A random translation would give 5 secondary pentagons, i.e. 25 points. Here we have degeneracies due to ‘coinciding points’.

Cartoon version of a virus or carbon onion. Would there be an evolutionary beneﬁt to have more than just compact symmetry? The problem has an intrinsic length scale.

2D and 3D point arrays for applications to viruses, fullerenes, quasicrystals, proteins etc Two complementary ways to construct these

ISSN 2053-2733 Acta Crystallographica Section A

September 2013 Volume 54 Number 9 Volume 70 Foundations and Part 2 Advances

Journal of March 2014 Editors: S. J. L. Billinge and J. Miao Mathematical Physics

jmp.aip.org journals.iucr.org International Union of Crystallography Wiley-Blackwell

There are speciﬁc interactions between RNA and coat protein (CP) given by symmetry axes Essential for assembly as only this RNA-CP interaction turns CP into right geometric shape for capsid formation The RNA forms a Hamiltonian cycle visiting each CP once – dictated by symmetry A patent for a new antiviral strategy (Reidun Twarock)

Extend idea of affine symmetry to other icosahedral objects in nature: football-shaped fullerenes Recover different shells with icosahedral symmetry from affine approach: carbon onions( C60 − C240 − C540)

Get nested arrangements like Russian dolls: carbon onions (e.g. Nature 510, 250253) Potential to extend to other known carbon onions with diﬀerent start conﬁguration, chirality etc

ISSN 2053-2733 Acta Crystallographica Section A

Volume 70 Foundations and Part 2 Advances

March 2014 Editors: S. J. L. Billinge and J. Miao

journals.iucr.org International Union of Crystallography Wiley-Blackwell

Non-compact symmetry that relates diﬀerent structural features in the same polyhedral object Novel symmetry principle in Nature, shown that it seems to apply to at least fullerenes and viruses

Pierre-Philippe Dechant Root systems & Cliﬀord algebras: from symmetries of viruses to E8 & an ADE correspondence Overview

1 Viruses, root systems and aﬃne extensions Viruses Root systems Aﬃne extensions

2 Cliﬀord algebras and exceptional root systems Cliﬀord basics E8 from the icosahedron 3D to 4D spinor induction Trinities and McKay correspondence

3 The Coxeter plane Clifford basics Viruses, root systems and affine extensions E from the icosahedron Clifford algebras and exceptional root systems 8 3D to 4D spinor induction The Coxeter plane Trinities and McKay correspondence Clifford Algebra and orthogonal transformations

Form an algebra using the product between two vectors

ab ≡ a · b + a ∧ b

1 Inner product is symmetric part a · b = 2 (ab + ba) Reflecting x in n is given by x0 = x − 2(x · n)n = −nxn (n and −n doubly cover the same reflection) Via Cartan-Dieudonné theorem any orthogonal (/conformal/modular) transformation can be written as successive reflections

0 x = ±n1n2 ...nk xnk ...n2n1 = ±AxA˜

Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Clifford basics Viruses, root systems and affine extensions E from the icosahedron Clifford algebras and exceptional root systems 8 3D to 4D spinor induction The Coxeter plane Trinities and McKay correspondence Clifford Algebra of 3D

E.g. Pauli algebra in 3D (likewise for Dirac algebra in 4D) is

{1} {e1,e2,e3} {e1e2,e2e3,e3e1} {I ≡ e1e2e3} |{z} | {z } | {z } | {z } 1 scalar 3 vectors 3 bivectors 1 trivector

We can multiply together root vectors in this algebra αi αj ... A general element has8 components, even products (rotations/spinors) have four components: ˜ 2 2 2 2 R = a0 + a1e2e3 + a2e3e1 + a3e1e2 ⇒ RR = a0 + a1 + a2 + a3 So behaves as a 4D Euclidean object – inner product 1 (R ,R ) = (R R˜ + R R˜ ) 1 2 2 2 1 1 2

Exceptional E8 (projected into the Coxeter plane)

E8 root system has 240 roots, H3 has order 120

Order 120 group H3 doubly covered by 240 (s)pinors in 8D space

With (somewhat counterintuitive) reduced inner product this gives the E8 root system

E8 is actually hidden within 3D geometry!

α4

α1 α2 α3 τα4 τα3 τα2 τα1

The 6 roots( ±1,0,0) and permutations of A1 × A1 × A1 generate 8 spinors:

±e1, ±e2, ±e3 give the 8 spinors ±1,±e1e2,±e2e3,±e3e1 This is a discrete spinor group isomorphic to the quaternion group Q. As 4D vectors these are (±1,0,0,0) and permutations, the 8 roots of A1 × A1 × A1 × A1 (the 16-cell).

Pierre-Philippe Dechant Root systems & Cliﬀord algebras: from symmetries of viruses to E8 & an ADE correspondence Check axioms: 1. Φ ∩ Rα = {−α,α} ∀ α ∈ Φ

2. sα Φ = Φ ∀ α ∈ Φ Proof: 1. R and −R are in a spinor group by construction (double cover of orthogonal transformations), 2. closure under reﬂections is guaranteed by the closure property of the spinor group (with a twist: −R1R˜2R1)

Clifford basics Viruses, root systems and affine extensions E from the icosahedron Clifford algebras and exceptional root systems 8 3D to 4D spinor induction The Coxeter plane Trinities and McKay correspondence Induction Theorem – root systems

Induction Theorem: every 3D root system gives a 3D spinor group which gives a 4D root system.

Pierre-Philippe Dechant Root systems & Cliﬀord algebras: from symmetries of viruses to E8 & an ADE correspondence Proof: 1. R and −R are in a spinor group by construction (double cover of orthogonal transformations), 2. closure under reﬂections is guaranteed by the closure property of the spinor group (with a twist: −R1R˜2R1)

Induction Theorem: every 3D root system gives a 3D spinor group which gives a 4D root system. Check axioms: 1. Φ ∩ Rα = {−α,α} ∀ α ∈ Φ

2. sα Φ = Φ ∀ α ∈ Φ

Induction Theorem: every 3D root system gives a 3D spinor group which gives a 4D root system. Check axioms: 1. Φ ∩ Rα = {−α,α} ∀ α ∈ Φ

H4 from H3

The H3 root system has 30 roots e.g. simple roots 1 α = e ,α = − ((τ − 1)e + e + τe ) and α = e . 1 2 2 2 1 2 3 3 3 Subgroup of rotations A5 of order 60 is doubly covered by 120 1 spinors of the form α α = − (1 − (τ − 1)e e + τe e ) , 1 2 2 1 2 2 3 1 α α = e e and α α = − (τ − (τ − 1)e e + e e ) . 1 3 2 3 2 3 2 3 1 2 3

1 (±1,0,0,0) (8 perms) , (±1,±1,±1,±1) (16 perms) 2 1 (0,±1,±σ,±τ) (96 even perms), 2 As 4D vectors are the 120 roots of the H4 root system.

The 3D Coxeter groups that are symmetry groups of the Platonic Solids:

The 6/12/18/30 roots in A1 × A1 × A1/A3/B3/H3 generate 8/24/48/120 spinors.

E.g. ±e1, ±e2, ±e3 give the 8 spinors ±1,±e1e2,±e2e3,±e3e1 The discrete spinor group is isomorphic to the quaternion group Q / binary tetrahedral group2 T / binary octahedral group2 O/ binary icosahedral group2 I ).

Exceptional phenomena: D4 (triality, important in string theory), F4 (largest lattice symmetry in 4D), H4 (largest non-crystallographic symmetry); Exceptional D4 and F4 arise from series A3 and B3

rank-3 group diagram binary rank-4 group diagram

A1 × A1 × A1 Q A1 × A1 × A1 × A1

A3 2T D4 4 4 B3 2O F4 5 5 H3 2I H4

Arnold’s observation that many areas of real mathematics can be complexified and quaternionified resulting in theories with a similar structure. The fundamental trinity is thus (R,C,H) n n n The projective spaces( RP ,CP ,HP ) 1 1 1 2 1 4 The spheres( RP = S ,CP = S ,HP = S ) The Möbius/Hopfbundles( S1 → S1,S3 → S2,S7 → S4)

The Lie Algebras( E6,E7,E8) The symmetries of the Platonic Solids( A3,B3,H3) The 4D groups( D4,F4,H4) New connections via my Cliﬀord spinor construction (see McKay correspondence)

Arnold’s connection between (A3,B3,H3) and (D4,F4,H4) is very convoluted and involves numerous other trinities at intermediate steps: Decomposition of the projective plane into Weyl chambers and Springer cones The number of Weyl chambers in each segment is 24 = 2(1 + 3 + 3 + 5),48 = 2(1 + 5 + 7 + 11),120 = 2(1 + 11 + 19 + 29) Notice this miraculously is one less than the degrees of invariants ((2,4,4,6),(2,6,8,12),(2,12,20,30)) of the Coxeter groups (D4,F4,H4) Believe the Cliﬀord connection is more direct

Group Discrete subgroup Action Mechanism SO(3) rotational (chiral) x → RxR˜ O(3) reﬂection (full/Coxeter) x → ±AxA˜ Spin(3) binary (R1,R2) → R1R2 Pin(3) pinor (A1,A2) → A1A2

e.g. the chiral icosahedral group has 60 elements, encoded in Cliﬀord by 120 spinors, which form the binary icosahedral group together with the inversion/pseudoscalar I this gives 60 rotations and 60 rotoinversions, i.e. the full icosahedral group H3 in 120 elements (with 240 pinors)

Easy enough to calculate conjugacy classes etc of pinors in Cliﬀord algebra Chiral (binary) polyhedral groups have irreps 0 00 0 00 tetrahedral (12/24): 1, 1 , 1 ,2 s ,2 s ,2 s , 3 0 0 0 octahedral (24/48): 1, 1 , 2,2 s ,2 s , 3, 3 ,4 s 0 ¯ icosahedral (60/120): 1,2 s ,2 s , 3, 3, 4,4 s , 5,6 s Binary groups are discrete subgroups of SU(2) and all thus have a2 s spinor irrep Connection with the McKay correspondence!

More than E-type groups: the inﬁnite family of 2D groups, the binary cyclic and dicyclic groups are in correspondence with An and + Dn, e.g. the quaternion group Q and D4 . So McKay correspondence not just a trinity but ADE-classiﬁcation. We also have I2(n) on top of the trinity (A3,B3,H3)

Pierre-Philippe Dechant Root systems & Cliﬀord algebras: from symmetries of viruses to E8 & an ADE correspondence Overview

1 Viruses, root systems and aﬃne extensions Viruses Root systems Aﬃne extensions

2 Cliﬀord algebras and exceptional root systems Cliﬀord basics E8 from the icosahedron 3D to 4D spinor induction Trinities and McKay correspondence

3 The Coxeter plane Viruses, root systems and aﬃne extensions Cliﬀord algebras and exceptional root systems The Coxeter plane The Coxeter Plane

Every (for our purposes) Coxeter group has a Coxeter plane. A way to visualise Coxeter groups in any dimension by projecting their root system onto the Coxeter plane

Like the symmetric group, Coxeter groups can have invariant polynomials. Their degrees d are important invariants/group characteristics. Turns out that actually degrees d are intimately related to so-called exponents mm = d − 1 .

A Coxeter Element is any combination of all the simple reﬂections w = s1 ...sn , i.e. in Cliﬀord algebra it is encoded

by the versor W = α1 ...αn acting as v → wv = ±W˜ vW . All such elements are conjugate and thus their order is invariant and called the Coxeter number h. The Coxeter element has complex eigenvalues of the form exp(2πmi/h) where m are called exponents: wx = exp(2πmi/h)x Standard theory complexiﬁes the real Coxeter group situation in order to ﬁnd complex eigenvalues, then takes real sections again.

The Coxeter element has complex eigenvalues of the form exp(2πmi/h) where m are called exponents Standard theory complexiﬁes the real Coxeter group situation in order to ﬁnd complex eigenvalues, then takes real sections again. In particular,1 and h − 1 are always exponents Turns out that actually exponents and degrees are intimately related ( m = d − 1 ). The construction is slightly roundabout but uniform, and uses the Coxeter plane.

In particular, can show every (for our purposes) Coxeter group has a Coxeter plane Existence relies on the fact that all groups in question have tree-like Dynkin diagrams, and thus admit an alternate colouring Essentially just gives two sets of mutually commuting generators

Existence relies on the fact that all groups in question have tree-like Dynkin diagrams, and thus admit an alternate colouring Essentially just gives two sets of orthogonal = mutually commuting generators but anticommuting root vectors αw and αb (duals ω) Cartan matrices are positive deﬁnite, and thus have a Perron-Frobenius (all positive) eigenvector λi . Take linear combinations of components of this eigenvector as coeﬃcients of two vectors from the orthogonal sets vw = ∑λw ωw and vb = ∑λbωb Their outer product/Coxeter plane bivector BC = vb ∧ vw describes an invariant plane where w acts by rotation by2 π/h.

n I2(n) π π For I2(n) take α1 = e1, α2 = −cos n e1 + sin n e2 So Coxeter versor is just π π πI W = α α = −cos + sin e e = −exp − 1 2 n n 1 2 n In Cliﬀord algebra it is therefore immediately obvious that the action of the I2(n) Coxeter element is described by a versor (here a rotor/spinor) that encodes rotations in the e1e2-Coxeter-plane and yields h = n since trivially W n = (−1)n+1 yielding w n = 1 via wv = W˜ vW .

πI So Coxeter versor is just W = −exp − n

I = e1e2 anticommutes with both e1 and e2 such that sandwiching formula becomes 2πI v → wv = W˜ vW = W˜ 2v = exp ± v immediately n yielding the standard result for the complex eigenvalues in real Clifford algebra without any need for artificial complexification

The Coxeter plane bivector BC = e1e2 = I gives the complex structure

The Coxeter plane bivector BC is invariant under the Coxeter versor WB˜ C W = ±BC .

Turns out in Cliﬀord algebra we can factorise W into orthogonal (commuting/anticommuting) components

W = α1 ...αn = W1 ...Wn with Wi = exp(πmi Ii /h)

2 Here, Ii is a bivector describing a plane with Ii = −1

For v orthogonal to the plane descrbed by Ii we have

v → W˜ i vWi = W˜ i Wi v = v so cancels out For v in the plane we have ˜ ˜ 2 v → Wi vWi = Wi v = exp(2πmi Ii /h)v Thus if we decompose W into orthogonal eigenspaces, in the eigenvector equation all orthogonal bits cancel out and one gets the complex eigenvalue from the respective eigenspace

For v in the plane we have ˜ ˜ 2 v → Wi vWi = Wi v = exp(2πmi Ii /h)v So complex eigenvalue equation arises geometrically without any need for complexification Different complex structures immediately give different eigenplanes Eigenvalues/angles/exponents given from just factorising W = α1 ...αn E.g. H4 has exponents 1,11,19,29 and π 11π W = exp 30 BC exp 30 IBC Here we have been looking for orthogonal eigenspaces, so innocuous – different complex structures commute But not in general – naive complexification can be misleading

4D case: D4

E.g. D4 has exponents 1,3,3,5 Coxeter versor decomposes into orthogonal components π 3π W = α α α α = exp B exp IB 1 2 3 4 6 C 6 C

4D case: F4

E.g. F4 has exponents 1,5,7,11 Coxeter versor decomposes into orthogonal components π 5π W = α α α α = exp B exp IB 1 2 3 4 12 C 12 C

4D case: H4

E.g. H4 has exponents 1,11,19,29 Coxeter versor decomposes into orthogonal components π 11π W = α α α α = exp B exp IB 1 2 3 4 30 C 30 C

rank 4 exponents W-factorisation π 2π A4 1,2,3,4 W = exp 5 BC exp 5 IBC π 3π B4 1,3,5,7 W = exp 8 BC exp 8 IBC π π D4 1,3,3,5 W = exp 6 BC exp 2 IBC π 5π F4 1,5,7,11 W = exp 12 BC exp 12 IBC π 11π H4 1,11,19,29 W = exp 30 BC exp 30 IBC

8D case: E8

E.g. H4 has exponents 1,11,19,29, E8 has 1,7,11,13,17,19,23,29 Coxeter versor decomposes into orthogonal components π 7π 11π 13π W = α ...α = exp( B )exp( B )exp( B )exp( B ) 1 8 30 C 30 2 30 3 30 4

8D case: E8

So what has been gained by this Clifford view? There are different entities that serve as unit imaginaries They have a geometric interpretation as an eigenplane of the Coxeter element These don’t need to commute with everything like i (though they do here – at least anticommute. But that is because we looked for orthogonal decompositions) But see that in general naive complexification can be a dangerous thing to do – unnecessary, issues of commutativity, confusing different imaginaries etc

The countably inﬁnite family I2(n) and ADE

3 For A1 can see immediately 8 = 2(1 + 1 + 1 + 1) Simple roots α1 = e1, α2 = e2, α3 = e3, α4 = e4 give π π π π W = e1e2e3e4 = (cos 2 + sin 2 e1e2)(cos 2 + sin 2 e3e4) = π π exp( 2 e1e2)exp( 2 e3e4) Gives exponents (1,1,1,1) (from h − 1 = 2 − 1)

The countably inﬁnite family I2(n) and ADE

For A1 × I2(n) one gets the same decomposition 4n = 2(1 + (n − 1) + 1 + (n − 1)) = 2 · 2n π π Simple roots α1 = e1, α2 = −cos n e1 + sin n e2, α3 = e3, π π πe1e2 πe3e4 α4 = −cos n e3 + sin n e4 give W = exp − n exp − n Gives exponents (1,(n − 1),1,(n − 1))

The countably inﬁnite family I2(n) and ADE

So Arnold’s initial hunch regarding the exponents extends in fact to my full correspondence McKay correspondence is a correspondence between even subgroups of SU(2)/quaternions and ADE aﬃne Lie algebras In fact here get the even quaternion subgroups from 3D – link to ADE aﬃne Lie algebras via McKay?

The countably inﬁnite family I2(n), ﬁnite subgroups of SU(2) and ADE

Double cover of cyclic group of order n (binary cyclic group is I2(n) as a root system – corresponding to An aﬃne Lie algebras (2D root systems are self-dual – just see complex part) The dicyclic group/binary dihedral group is I2(n) × I2(n)– corresponding to Dn aﬃne Lie algebras The binary tetrahedral, octahedral and icosahedral groups are D4, F4 and H4 as root systems – and correspond to E-type Lie algebras E6, E7, E8 Get all of the even subgroups of the quaternions from 2D and 3D root systems and via McKay correspondence the ADE Lie algebras

(I2(n),A1 × I2(n),A3,B3,H3) give 4D root systems (I2(n),I2(n) × I2(n),D4,F4,H4) These are in McKay correspondence with ADE Lie algebras (A2n−1,Dn+2,E6,E7,E8) The numbers of roots (2n,2n + 2,12,18,30) is also the sum of the dimensions of the irreps and the ADE Coxeter number Is there a direct correspondence between the Platonic root systems and the ADE root systems?

2D/3D root systems (I2(n),A1 × I2(n),A3,B3,H3) 2D/4D root systems (I2(n),I2(n) × I2(n),D4,F4,H4) ADE root systems (An,Dn+2,E6,E7,E8)

All exceptional geometries arise in 3D, root systems giving rise to Lie groups/algebras etc Completely novel spinorial way of viewing exceptional geometries as 3D phenomena – implications for HEP etc? New view of Coxeter degrees and exponents with geometric interpretation of imaginaries A uniﬁed framework for doing group and representation theory: polyhedral, orthogonal, conformal, modular (Moonshine) etc Correspondence between 2D/3D & 2D/4D root systems extends to full ADE correspondence with McKay and generalising Arnold’s link between trinities via 3D Weyl chamber decomposition and exponents in the 4D Coxeter plane

Thank you!

α8

α1 α2 α3 α4 α5 α6 α7

sβ1 = sα1 sα7 , sβ2 = sα2 sα6 , sβ3 = sα3 sα5 , sβ4 = sα4 sα8 ⇒ H4

E8 has a H4 subgroup of rotations via a ‘partial folding’ Can project 240 E8 roots to H4 + τH4 – essentially the reverse of the previous construction!

Coxeter element & number of E8 and H4 are the same

4D root systems are surprisingly relevant to HEP

A4 is SU(5) and comes up in Grand Uniﬁcation

D4 is SO(8) and is the little group of String theory In particular, its triality symmetry is crucial for showing the equivalence of RNS and GS strings

B4 is SO(9) and is the little group of M-Theory

F4 is the largest crystallographic symmetry in 4D and H4 is the largest non-crystallographic group The above are subgroups of the latter two Spinorial nature of the root systems could have surprising consequences for HEP

Pierre-Philippe Dechant Root systems & Cliﬀord algebras: from symmetries of viruses to E8 & an ADE correspondence